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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution</dfn> Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_11.html">
\begin{equation}
({\bf A}-r{\bf I}) \vec{\xi}={\bf 0}.\tag{6.4.5}
\end{equation}
</div>
<p class="continuation">We require</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_11.html">
\begin{equation*}
|{\bf A}-r {\bf I}|=0 \to 
\left|
\begin{array}{cc}
4-r &amp; -2\\
8 &amp; -4-r
\end{array}
\right|=0 \to r^2=0 \to r_1=r_2=0.
\end{equation*}
</div>
<p class="continuation">For <span class="process-math">\(r_1=r_2=0\text{,}\)</span> from (<a href="" class="xref" data-knowl="./knowl/eq7_11.html" title="Equation 6.4.5">(6.4.5)</a>),</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_11.html">
\begin{equation}
\left(
\begin{array}{cc}
4 &amp; -2\\
8 &amp; -4
\end{array}
\right) 
\left(
\begin{array}{c}
\xi^{(1)}_1\\
\xi^{(1)}_2
\end{array}
\right)={\bf 0} \to 
\begin{array}{c}
4 \xi^{(1)}_1-2 \xi^{(1)}_2=0\\
8 \xi^{(1)}_1-4\xi^{(1)}_2=0
\end{array}
\to \xi^{(1)}_2=2 \xi^{(1)}_1.\tag{6.4.6}
\end{equation}
</div>
<p class="continuation">Choose <span class="process-math">\(\xi^{(1)}_1=1\text{,}\)</span> the eigenvector is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_11.html">
\begin{equation*}
\vec{\xi}^{(1)}=\left(
\begin{array}{c}
1 \\
2
\end{array}
\right)
\end{equation*}
</div>
<p class="continuation">and one solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_11.html">
\begin{equation*}
{\bf x}^{(1)}=\vec{\xi}^{(1)} e^{r_1 t}=\left(
\begin{array}{c}
1 \\
2
\end{array}
\right).
\end{equation*}
</div>
<span class="incontext"><a href="sec6_4.html#p-273" class="internal">in-context</a></span>
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